K-density, N-density, and finiteness properties

Abstract

Modelling non-sequential processes by partially ordered sets (posets) leads to the concept of K-density which says that every cut and every line have (exactly) one point in common. The “simplest” example of non-K-density is given by a four-element poset the underlying graph of wich is “N-shaped”; a poset is called N-dense iff every (four-element) N-shaped subposet can be extended to an K-dense subposet by addition of one point. K-density implies N-density; for finite non-empty posets also the converse implication is true. It turns out that much weaker properties are sufficient; especially, it will be proved that an N-dense non-empty poset is K-dense if all cuts are finite.